Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.


Introduction and Main Results
Let T be the classical singular integral operator. In 1976, Coifman, Rochberg and Weiss [4] studied the commutator generated by T and a function b ∈ BM O(R n ) as follows (1.1) and sufficient conditions for the boundedness of [b, M ] in L p (R n ) when b belongs to BM O(R n ).
In 2009, Zhang and Wu [28] extended their results to commutators of the fractional maximal function. The results in [2] and [28] were extended to variable Lebesgue spaces in [29] and [30].
Recently, Zhang [26] studied the commutator [b, M ] when b belongs to Lipschitz spaces.
Some necessary and sufficient conditions for the boundedness of [b, M ] on Lebesgue and Morrey spaces are given. Some of the results were extended to variable Lebesgue spaces in [27] and to the context of Orlicz spaces in [15], [16] and [31].
Motivated by the papers mentioned above, in this paper, we mainly study the mapping also give affirmative answers to the questions mentioned in [16] and [29] (see Remark 1.4 and Remark 1.5 below, respectively). We would like to note that some of our results are new even in the case of Lebesgue spaces with constant exponents.
To state the results, we first recall some definitions and notations.
Let γ ≥ 0, for a fixed cube Q 0 , the fractional maximal function with respect to Q 0 of a locally integrable function f is given by where the supremum is taken over all cubes Q such that x ∈ Q ⊆ Q 0 .
When γ = 0, we simply write M Q 0 instead of M 0,Q 0 . Definition 1.2 Let 0 < β < 1, we say a function b belongs to the Lipschitz spaceΛ β (R n ), The smallest such constant C is called theΛ β norm of b and is denoted by b Λ β .
where the supremum is taken over all cubes Q in R n .
For a function b defined on R n , we denote by The set L p(·) (R n ) becomes a Banach space with respect to the norm We refer to [5], [10], [21] and [22] for more details on function spaces with variable exponents.
Denote by P(R n ) the set of all measurable functions p(·) : R n → [1, ∞) such that 1 < p − := ess inf x∈R n p(x) and p + : = ess sup and by B(R n ) the set of all p(·) ∈ P(R n ) such that M is bounded on L p(·) (R n ).
For notational convenience, we introduce a notation B γ (R n ) as follows.
Our results can be stated as follows.
(4) There exists s(·) ∈ B(R n ) such that For the case p(·) and q(·) being constants, we have the following results from Theorem 1.1, which is new even for this case.
For the case p(·) and q(·) being constants, we have the following results by Theorem 1.2.
Corollary 1.2 Let 0 < α < n and b be a locally integrable function. Then the following statements are equivalent: for some p and q such that 1 < p < n/α and for all p and q such that 1 < p < n/α and Remark 1.6 It was shown in [2] and [28] that statements (1), (2) and (3) are equivalent to respectively. Compared with (1.7), (1.6) gives a new characterization.
Next, we give some necessary and sufficient conditions for the boundedness of the maximal commutator M α,b on variable Lebegue spaces when b belongs to Lipschitz space. Theorem 1.3 Let 0 < β < 1, 0 < α < n, 0 < α + β < n and b be a locally integrable function. Then the following statements are equivalent: (4) There exists s(·) ∈ B(R n ) such that Remark 1.7 For the case α = 0, similar results were given in [26] for Lebesgue spaces with constant exponents and in [27] for the variable case.
Finally, for the case of completeness of this paper, we state a result similar to Theorem 1.3 without proof, which can be deduced from [29] and [18].
If p(·) and q(·) are constants, we have a result similar to Corollary 1.3. We omit the details.
The remainder of this paper is organized as follows. In the next section, we give some lemmas that will be used later. In Section 3, we prove Theorems 1.1, 1.2 and 1.3.
It is known that Lipschitz spaceΛ β (R n ) coincides with some Morrey-Companato space (see, e.g., [20]) and can be characterized by mean oscillation as the following lemma, which is due to DeVore and Sharpley [9] and Janson, Taibleson and Weiss [20] (see also Paluszyński [24]).
From the proof of Theorem 1.4 in [26], we can obtain the following characterization of nonnegative Lipschitz functions.
Lemma 2.2 Let 0 < β < 1 and b be a locally integrable function. Then the following statements are equivalent: (2) For all 1 ≤ s < ∞, (2) There exists s ∈ [1, ∞) such that The following strong-type estimates for the fractional maximal function is well known, see [11] or [14] for details.
Lemma 2.4 Let 0 < γ < n, 1 < p < n/γ and 1/q = 1/p − γ/n. Then there exists a positive constant C(n, γ, p) such that As for the boundedness of the fractional maximal function on variable Lebesgue spaces, the following result was given in [6]. See Corollary 2.12 and Remark 2.13 in [6] for details.
We also need some basic properties of variable Lebesgue spaces. Denoted by p ′ (·) the conjugate index of p(·). Obviously, if p(·) ∈ P(R n ) then p(·) ∈ P(R n ). The following lemma is known as the generalized Hölder's inequality in variable Lebesgue spaces. See [5] and [10] for details.

Lemma 2.8 ([7])
Given p(·) ∈ P(R n ), then for all r > 0 we have for all cubes Q in R n .
If q(·)(n − γ)/n ∈ B(R n ), then there exists a constant C > 0 such that for all cubes Q in R n . Now, we give the following pointwise estimates for [b, M α ] when b ∈Λ β (R n ).

Lemma 2.12 ([2]
, [28]) Let 0 ≤ γ < n, Q be a cube in R n and f be a locally integrable function. Then for all x ∈ Q,

Proofs of Theorems 1.1, 1.2 and 1.3
To prove Theorem 1.1, we first prove the following lemma.
Lemma 3.1 Let 0 < β < 1 and 0 < γ < n. If b is a locally integrable function and satisfies Proof Some ideas are taken from [2], [28] and [29]. Reasoning as the proof of (4.4) in [29], see also the proof of Lemma 2.4 in [28], we have, for any cube Q, Indeed, for any cube It is easy to check that the following equality is true (see [2] page 3331): Noticing the obvious estimate Then, for any x ∈ E, Therefore, By Lemma 2.7 (i), (3.1) and Lemma 2.9, we get So, the proof is completed by applying Lemma 2.1. ( to L q(·) (R n ) for all p(·), q(·) ∈ B α+β (R n ). For such p(·) and any f ∈ L p(·) (R n ), it follow from Lemma 2.6 that M α (f )(x) < ∞ for almost every x ∈ R n . By Lemma 2.11 we have Then, statement (3) follows from Lemma 2.5.
(4) ⇒ (1). By Lemma 2.2, it suffices to prove For any fixed cube Q, (3.3) For I 1 , by statement (4) and applying Lemma 2.7 (i) and Lemma 2.9, we have where the constant C is independent of Q.
Next, we consider I 2 . Similar to the ones in the proof of Theorem 1.1 in [31], we can get I 2 ≤ C. Now, we give the proof of it. For all x ∈ Q, it follows from Lemma 2.12 that Then, for any x ∈ Q, Since s(·) ∈ B(R n ), then statement (4) along with Lemma 3.1 gives b ∈Λ β (R n ), which implies |b| ∈Λ β (R n ). Thus, we can apply Lemma 2.11 to [|b|, M α ] and [|b|, M ] since |b| ∈Λ β (R n ) and |b| ≥ 0. By Lemmas 2.11 and 2.12 we have, for any x ∈ Q,

By (3.4) we have
Putting the above estimates for I 1 and I 2 into (3.3) we obtain (3.2).
Noting that 1 it follows from Lemma 2.7 (ii), (3.5) and Lemma 2.8 that which is what we want.
The proof of Theorem 1.1 is finished.
To prove Theorem 1.2, we recall the following results obtained in [29].
Lemma 3.3 Let 0 < γ < n. If b is a locally integrable function and satisfies for some s(·) ∈ B(R n ), then b ∈ BM O(R n ).
Proof of Theorem 1.2 Since the equivalence of (1), (2) and (3)  For any fixed cube Q, it follows from (3.3) and (3.4) that (3.6) For J 1 , by Lemma 2.7 (i), Lemma 2.9 and statement (4) we have where the constant C is independent of Q.
Then, it follows from Lemma 2.7 (i), Lemma 2.10 and Lemma 2.9 that Similarly, by Lemma 2.7 (i) and Lemma 2.9, we have Putting the above estimates for J 1 , J 2 and J 3 into (3.6), we obtain which implies b ∈ BM O(R n ) and b − ∈ L ∞ (R n ) by Lemma 2.3, since the constant C is independent of Q.
The proof of Theorem 1.2 is completed.
The proof of Theorem 1.3 is finished.